Applied Numerical Analysis & Computational Mathematics最新文献

A summary approach in the determination of eigenvalues and eigenvectors of transport of information fields in 2-port cell networks, is presented. The dispersion relation whose nodes constitute the eigenvalues is formed as a sum of diagrams which topologically are mapped 1-1 to the graphs of our paper. We find the matrix representation of these graphs, as an Abelian semigroup of the Boolean matrices formed by all the possible graphs of this kind. We also find the algebra that distinguishes the class of these matrices from the rest of the general Boolean matrices. Generalization of these symmetry properties for n-port cell finite networks is currently being investigated. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, we propose product-type iterative methods with restart for avoiding breakdown. The residual vectors of product-type iterative methods are denoted as the product of a polynomial and the residual vector of BiCG. In particular, the residual polynomial of BiCG is called the Lanczos polynomial. In product-type iterative methods, breakdown may occur in the recurrence relations of the Lanczos polynomial due to division by zero. We investigated the behavior of the proposed method through experiments using Toeplitz linear systems with various parameters. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The paper presents an explicit finite-difference method for the numerical solution of the Sine-Gordon equation in two space variables, as it arises, for example, in rectangular large-area Josephson junction. The dispersive nonlinear partial differential equation of the system allows for soliton-type solutions, an ubiquitous phenomenon in a large-variety of physical problems.

The method, which is based on fourth order rational approximants of the matrix-exponential term in a three-time level recurrence relation, after the application of finite-difference approximations, it leads finally to a second order initial value problem. Because of the existing sinus term this problem becomes nonlinear. To avoid solving the arising nonlinear system a new method based on a predictor-corrector scheme is applied. Both the nonlinear method and the predictor-corrector are analyzed for local truncation error, stability and convergence. Numerical solutions for cases involving the most known from the bibliography ring and line solitons are given. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We present a novel representation for generalized hypergeometric functions of type _p+1F_p which is in fact defined by an infinite series in nonnegative integer powers of its argument. We first construct a first order vector differential equation such that the unknown vector's coefficient is the sum of a constant matrix and a matrix premultiplied by the reciprocal of the independent variable whereas its first order derivative has unit matrix coefficient. An infinite process of factor extractions and power annihilations is employed yielding finally a vector differential equation that can be easily and analytically solved. Truncation of this scheme can be used to get approximations to hypergeometric functions of type _p+1F_p. These functions have regular singularities at 0 and 1 values of the independent variable together with another regular singularity at infinity. Hence the factors are chosen to reflect the expected behavior of the functions at the singular point in a descending contribution order. Factorization is realized also for regular points. A simple, yet meaningful, implementation seems to give quite promising results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Recently we have presented a matrix algebraic factorization scheme for multiplicative representations of generalized hypergeometric functions of type _p+1F_p. The Method uses exponential functions with matrix arguments. We have shown that factorization is possible around any kind of point, regular or singular, and the constant matrices appearing in the argument of the exponential functions. According to the theory of linear ordinary differential equations, a series solution constructed around a point converges in the disk centered at that point with a radius equal to the difference from that point to the nearest singularity of the differential equation under consideration. Although we do not use an additive series solution, it is not hard to show that the same convergence property is expected from the factorized solutions. This paper contains the construction of the matrices transforming one evolution matrix at a singular point to another. This is done for all singularities located at z = 0, z = 1 and infinity. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The numerical range of an n × n matrix polynomial P(λ) = A_mλ^m + A_m–1λ^m–1 + … + A₁λ + A₀ is defined by W(P) = {λ ∈ ℂ : x*P(λ)x = 0, x ∈ ℂⁿ, x*x = 1}, and plays an important role in the study of matrix polynomials. In this paper, we describe a methodology for the illustration of its boundary, ∂W(P), using recent theoretical results on numerical ranges and algebraic curves. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Two new semilocal convergence results of Newton-Kantorovich type are presented for the Halley method, where the usual convergence conditions, which appears in the literature, are relaxed. In one of them, it is supposed that the second derivative F″ of a nonlinear operator F satisfies ‖F″(x₀)‖ ≤ α instead of ‖F″(x)‖ ≤ M, for all x in a subset of the domain of F, where α and M are positive real constants. In the other one fewer convergence conditions are required than all the existing ones until now. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this work we have investigated the stability and robustness of the optimal control solutions to a quantum system when the control duration goes to zero. The solutions at this limit are called “Instantaneous Solutions”. These investigations are based on the second variation of the cost functional evaluated at control solution values when the first variations of wave and costate functions are related to the first variation of external field amplitude via control equations. This form of cost functional's second variation is purely quadratic in the first variation of the external field amplitude. Investigations are conducted for an illustrative model system, one dimensional quantum harmonic oscillator under linear dipole interaction, purely quadratic objective operator in position, and purely quadratic penalty operator in momentum. We have not found the stability operator's spectrum explicitly. Instead we have employed a bound analysis to understand the system's stability and robustness. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The numerical solution of an eigenvalue problem for a set of ODEs may be non-trivial when high accuracy is needed and the interval of the independent variable extends to infinity. In that case, efficient asymptotics are needed at infinity to produce the initial conditions for numerical integration. Here such asymptotics are found for a set of N coupled 1D Schrödinger like ODEs in r, 0 ≤ r < ∞. This is a generalization of the well known phase integral approximation used for N = 1. Calculations are performed for N = 2; the ODEs describe small vibrations of a single quantum vortex in a Bose–Einstein condensate, where a critical situation arises in the long-wavelength limit, k → 0. The calculations were aimed at clarifying certain discrepancies in theoretical results pertaining to this limit. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this contribution the application of two inverse interpolation methods over finite fields is studied. More specifically, we consider the Aitken and Neville inverse interpolation methods for a “shifted” discrete exponential function. The results indicate that the computational cost of finding the discrete logarithm through this approach remains high, however interesting features regarding the degree of the resulting interpolation polynomials are reported. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)